Calculus Examples

$(0.36, 0.15)$

Step 1

To find the $\sin (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(0.36, 0.15)$ , draw the triangle between the three points $(0, 0)$ , $(0.36, 0)$ , and $(0.36, 0.15)$ .

Opposite : $0.15$

Adjacent : $0.36$

Step 2

Find the hypotenuse using Pythagorean theorem $c = \sqrt{a^{2} + b^{2}}$ .

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Step 2.1

Raise $0.36$ to the power of $2$ .

$\sqrt{0.1296 + {(0.15)}^{2}}$

Step 2.2

Raise $0.15$ to the power of $2$ .

$\sqrt{0.1296 + 0.0225}$

Step 2.3

Add $0.1296$ and $0.0225$ .

$\sqrt{0.1521}$

Step 3

$\sin (θ) = \frac{Opposite}{Hypotenuse}$ therefore $\sin (θ) = \frac{0.15}{\sqrt{0.1521}}$ .

$\frac{0.15}{\sqrt{0.1521}}$

Step 4

Simplify $\sin (θ)$ .

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Step 4.1

Multiply $\frac{0.15}{\sqrt{0.1521}}$ by $\frac{\sqrt{0.1521}}{\sqrt{0.1521}}$ .

$\sin (θ) = \frac{0.15}{\sqrt{0.1521}} \cdot \frac{\sqrt{0.1521}}{\sqrt{0.1521}}$

Step 4.2

Combine and simplify the denominator.

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Step 4.2.1

Multiply $\frac{0.15}{\sqrt{0.1521}}$ by $\frac{\sqrt{0.1521}}{\sqrt{0.1521}}$ .

$\sin (θ) = \frac{0.15 \sqrt{0.1521}}{\sqrt{0.1521} \sqrt{0.1521}}$

Step 4.2.2

Raise $\sqrt{0.1521}$ to the power of $1$ .

$\sin (θ) = \frac{0.15 \sqrt{0.1521}}{\sqrt{0.1521} \sqrt{0.1521}}$

Step 4.2.3

Raise $\sqrt{0.1521}$ to the power of $1$ .

$\sin (θ) = \frac{0.15 \sqrt{0.1521}}{\sqrt{0.1521} \sqrt{0.1521}}$

Step 4.2.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sin (θ) = \frac{0.15 \sqrt{0.1521}}{{\sqrt{0.1521}}^{1 + 1}}$

Step 4.2.5

Add $1$ and $1$ .

$\sin (θ) = \frac{0.15 \sqrt{0.1521}}{{\sqrt{0.1521}}^{2}}$

Step 4.2.6

Rewrite ${\sqrt{0.1521}}^{2}$ as $0.1521$ .

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Step 4.2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{0.1521}$ as ${0.1521}^{\frac{1}{2}}$ .

$\sin (θ) = \frac{0.15 \sqrt{0.1521}}{{({0.1521}^{\frac{1}{2}})}^{2}}$

Step 4.2.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sin (θ) = \frac{0.15 \sqrt{0.1521}}{{0.1521}^{\frac{1}{2} \cdot 2}}$

Step 4.2.6.3

Combine $\frac{1}{2}$ and $2$ .

$\sin (θ) = \frac{0.15 \sqrt{0.1521}}{{0.1521}^{\frac{2}{2}}}$

Step 4.2.6.4

Cancel the common factor of $2$ .

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Step 4.2.6.4.1

Cancel the common factor.

$\sin (θ) = \frac{0.15 \sqrt{0.1521}}{{0.1521}^{\frac{2}{2}}}$

Step 4.2.6.4.2

Rewrite the expression.

$\sin (θ) = \frac{0.15 \sqrt{0.1521}}{0.1521}$

Step 4.2.6.5

Evaluate the exponent.

$\sin (θ) = \frac{0.15 \sqrt{0.1521}}{0.1521}$

Step 4.3

Multiply $0.15$ by $\sqrt{0.1521}$ .

$\sin (θ) = \frac{0.0585}{0.1521}$

Step 4.4

Divide $0.0585$ by $0.1521$ .

$\sin (θ) = 0.\overline{384615}$